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Theorem isngp 21241
 Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isngp.n
isngp.z
isngp.d
Assertion
Ref Expression
isngp NrmGrp

Proof of Theorem isngp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3683 . . 3
21anbi1i 695 . 2
3 fveq2 5872 . . . . . 6
4 isngp.n . . . . . 6
53, 4syl6eqr 2516 . . . . 5
6 fveq2 5872 . . . . . 6
7 isngp.z . . . . . 6
86, 7syl6eqr 2516 . . . . 5
95, 8coeq12d 5177 . . . 4
10 fveq2 5872 . . . . 5
11 isngp.d . . . . 5
1210, 11syl6eqr 2516 . . . 4
139, 12sseq12d 3528 . . 3
14 df-ngp 21229 . . 3 NrmGrp
1513, 14elrab2 3259 . 2 NrmGrp
16 df-3an 975 . 2
172, 15, 163bitr4i 277 1 NrmGrp
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   w3a 973   wceq 1395   wcel 1819   cin 3470   wss 3471   ccom 5012  cfv 5594  cds 14720  cgrp 16179  csg 16181  cmt 20946  cnm 21222  NrmGrpcngp 21223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-co 5017  df-iota 5557  df-fv 5602  df-ngp 21229 This theorem is referenced by:  isngp2  21242  ngpgrp  21244  ngpms  21245  tngngp2  21291  cnngp  21412  zhmnrg  28101
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